Skip to main content

Abstract

Geometric finite elements (GFE) generalize the idea of Galerkin methods to variational problems for unknowns that map into nonlinear spaces. In particular, GFE methods introduce proper discrete function spaces that are conforming in the sense that values of geometric finite element functions are in the codomain manifold \(\mathcal {M}\) at any point. Several types of such spaces have been constructed, and some are even completely intrinsic, i.e., they can be defined without any surrounding space. GFE spaces enable the elegant numerical treatment of variational problems posed in Sobolev spaces with nonlinear codomain space. Indeed, as GFE spaces are geometrically conforming, such variational problems have natural formulations in GFE spaces. These correspond to the discrete formulations of classical finite element methods. Also, the canonical projection onto the discrete maps commutes with the differential for a suitable notion of the tangent bundle as a manifold, and we therefore also obtain natural weak formulations. Rigorous results exist that show the optimal behavior of the a priori L 2 and H 1 errors under reasonable smoothness assumptions. Although the discrete function spaces are no vector spaces, their elements can nevertheless be described by sets of coefficients, which live in the codomain manifold. Variational discrete problems can then be reformulated as algebraic minimization problems on the set of coefficients. These algebraic problems can be solved by established methods of manifold optimization. This text will explain the construction of several types of GFE spaces, discuss the corresponding test function spaces, and sketch the a priori error theory. It will also show computations of the harmonic maps problem, and of two example problems from nanomagnetics and plate mechanics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Abatzoglou, T.J.: The minimum norm projection on C 2-manifolds in \(\mathbb {R}^n\). Trans. Am. Math. Soc. 243, 115–122 (1978)

    Google Scholar 

  2. Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)

    Book  MATH  Google Scholar 

  3. Absil, P.A., Mahony, R., Trumpf, J.: An extrinsic look at the Riemannian Hessian. In: Geometric Science of Information. Lecture Notes in Computer Science, vol. 8085, pp. 361–368. Springer, Berlin (2013)

    Google Scholar 

  4. Absil, P.A., Gousenbourger, P.Y., Striewski, P., Wirth, B.: Differentiable piecewise-Bézier surfaces on Riemannian manifolds. SIAM J. Imaging Sci. 9(4), 1788–1828 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alouges, F.: A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case. SIAM J. Numer. Anal. 34(5), 1708–1726 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alouges, F., Jaisson, P.: Convergence of a finite element discretization for the landau–lifshitz equations in micromagnetism. Math. Models Methods Appl. Sci. 16(2), 299–316 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ambrosio, L.: Metric space valued functions of bounded variation. Ann. Sc. Norm. Super. Pisa Cl. Sci. 17(3), 439–478 (1990)

    MathSciNet  MATH  Google Scholar 

  8. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Springer, Berlin (2006)

    MATH  Google Scholar 

  9. Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med. 56(2), 411–421 (2006)

    Article  Google Scholar 

  10. Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 29(1), 328–347 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bartels, S., Prohl, A.: Constraint preserving implicit finite element discretization of harmonic map flow into spheres. Math. Comput. 76(260), 1847–1859 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Baumgarte, T.W., Shapiro, S.L.: Numerical Relativity – Solving Einstein’s Equations on the Computer. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  13. Belavin, A., Polyakov, A.: Metastable states of two-dimensional isotropic ferromagnets. JETP Lett. 22(10), 245–247 (1975)

    Google Scholar 

  14. Bergmann, R., Laus, F., Persch, J., Steidl, G.: Processing manifold-valued images. SIAM News 50(8), 1,3 (2017)

    Google Scholar 

  15. Berndt, J., Boeckx, E., Nagy, P.T., Vanhecke, L.: Geodesics on the unit tangent bundle. Proc. R. Soc. Edinb. A Math. 133(06), 1209–1229 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bogdanov, A., Hubert, A.: Thermodynamically stable magnetic vortex states in magnetic crystals. J. Magn. Magn. Mater. 138, 255–269 (1994)

    Article  Google Scholar 

  17. Buss, S.R., Fillmore, J.P.: Spherical averages and applications to spherical splines and interpolation. ACM Trans. Graph. 20, 95–126 (2001)

    Article  Google Scholar 

  18. Cartan, E.: Groupes simples clos et ouverts et géométrie riemannienne. J. Math. Pures Appl. 8, 1–34 (1929)

    MATH  Google Scholar 

  19. Chiron, D.: On the definitions of Sobolev and BV spaces into singular spaces and the trace problem. Commun. Contemp. Math. 9(04), 473–513 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Elsevier, Amsterdam (1978)

    MATH  Google Scholar 

  21. Convent, A., Van Schaftingen, J.: Intrinsic colocal weak derivatives and Sobolev spaces between manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. 16(5), 97–128 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Convent, A., Van Schaftingen, J.: Higher order weak differentiability and Sobolev spaces between manifolds (2017). arXiv preprint 1702.07171

    Google Scholar 

  23. de Gennes, P., Prost, J.: The Physics of Liquid Crystals. Clarendon Press, Oxford (1993)

    Google Scholar 

  24. Farin, G.: Curves and Surfaces for Computer Aided Geometric Design, 2nd edn. Academic, Boston (1990)

    MATH  Google Scholar 

  25. Fert, A., Reyren, N., Cros, V.: Magnetic skyrmions: advances in physics and potential applications. Nat. Rev. Mater. 2(17031) (2017)

    Google Scholar 

  26. Focardi, M., Spadaro, E.: An intrinsic approach to manifold constrained variational problems. Ann. Mat. Pura Appl. 192(1), 145–163 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Fréchet, M.: Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. Henri Poincaré 10(4), 215–310 (1948)

    MathSciNet  MATH  Google Scholar 

  28. Gawlik, E.S., Leok, M.: Embedding-based interpolation on the special orthogonal group. SIAM J. Sci. Comput. 40(2), A721–A746 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gawlik, E.S., Leok, M.: Interpolation on symmetric spaces via the generalized polar decomposition. Found. Comput. Math. 18(3), 757–788 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Giaquinta, M., Hildebrandt, S.: Calculus of Variations I. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2004). https://books.google.de/books?id=4NWZdMBH1fsC

  31. Grohs, P.: Quasi-interpolation in Riemannian manifolds. IMA J. Numer. Anal. 33(3), 849–874 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Grohs, P., Hardering, H., Sander, O.: Optimal a priori discretization error bounds for geodesic finite elements. Found. Comput. Math. 15(6), 1357–1411 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Grohs, P., Hardering, H., Sander, O., Sprecher, M.: Projection-based finite elements for nonlinear function spaces. SIAM J. Numer. Anal. 57(1), 404–428 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hajłasz, P.: Sobolev mappings between manifolds and metric spaces. In: Sobolev Spaces in Mathematics I. International Mathematical Series, vol. 8, pp. 185–222. Springer, Berlin (2009)

    Google Scholar 

  35. Hajlasz, P., Tyson, J.: Sobolev peano cubes. Michigan Math. J. 56(3), 687–702 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Hardering, H.: Intrinsic discretization error bounds for geodesic finite elements. Ph.D. thesis, Freie Universität Berlin (2015)

    Google Scholar 

  37. Hardering, H.: The Aubin–Nitsche trick for semilinear problems (2017). arXiv e-prints arXiv:1707.00963

    Google Scholar 

  38. Hardering, H.: L 2-discretization error bounds for maps into Riemannian manifolds (2018). ArXiv preprint 1612.06086

    Google Scholar 

  39. Hardering, H.: L 2-discretization error bounds for maps into Riemannian manifolds. Numer. Math. 139(2), 381–410 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  40. Hélein, F.: Harmonic Maps, Conservation Laws and Moving Frames, 2nd edn. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  41. Hélein, F., Wood, J.C.: Harmonic maps. In: Handbook of Global Analysis, pp. 417–491. Elsevier, Amsterdam (2008)

    Google Scholar 

  42. Jost, J.: Equilibrium maps between metric spaces. Calc. Var. Partial Differ. Equ. 2(2), 173–204 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  43. Jost, J.: Riemannian Geometry and Geometric Analysis, 6th edn. Springer, New York (2011)

    Book  MATH  Google Scholar 

  44. Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509–541 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ketov, S.V.: Quantum Non-linear Sigma-Models. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  46. Korevaar, N.J., Schoen, R.M.: Sobolev spaces and harmonic maps for metric space targets. Commun. Anal. Geom. 1(4), 561–659 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kowalski, O., Sekizawa, M.: Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles – a classification. Bull. Tokyo Gakugei Univ. 40, 1–29 (1997)

    MathSciNet  MATH  Google Scholar 

  48. Kružík, M., Prohl, A.: Recent developments in the modeling, analysis, and numerics of ferromagnetism. SIAM Rev. 48(3), 439–483 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  49. Lee, J.M.: Introduction to Smooth Manifolds. Springer, New York (2003)

    Book  Google Scholar 

  50. Melcher, C.: Chiral skyrmions in the plane. Proc. R. Soc. A 470(2172) (2014)

    Google Scholar 

  51. Mielke, A.: Finite elastoplasticity Lie groups and geodesics on SL(d). In: Newton, P., Holmes, P., Weinstein, A. (eds.) Geometry, Mechanics, and Dynamics, pp. 61–90. Springer, New York (2002)

    Chapter  MATH  Google Scholar 

  52. Münch, I.: Ein geometrisch und materiell nichtlineares Cosserat-Modell – Theorie, Numerik und Anwendungsmöglichkeiten

    Google Scholar 

  53. Reshetnyak, Y.G.: Sobolev classes of functions with values in a metric space. Sib. Mat. Zh. 38(3), 657–675 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  54. Rubin, M.: Cosserat Theories: Shells, Rods, and Points. Springer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  55. Sander, O.: Geodesic finite elements for Cosserat rods. Int. J. Numer. Methods Eng. 82(13), 1645–1670 (2010)

    MathSciNet  MATH  Google Scholar 

  56. Sander, O.: Geodesic finite elements on simplicial grids. Int. J. Numer. Methods Eng. 92(12), 999–1025 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  57. Sander, O.: Geodesic finite elements of higher order. IMA J. Numer. Anal. 36(1), 238–266 (2016)

    MathSciNet  MATH  Google Scholar 

  58. Sander, O.: Test function spaces for geometric finite elements (2016). ArXiv e-prints 1607.07479

    Google Scholar 

  59. Sander, O., Neff, P., Bîrsan, M.: Numerical treatment of a geometrically nonlinear planar Cosserat shell model. Comput. Mech. 57(5), 817–841 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  60. Shatah, J., Struwe, M.: Geometric Wave Equations. American Mathematical Society, Providence (2000)

    Book  MATH  Google Scholar 

  61. Simo, J., Fox, D., Rifai, M.: On a stress resultant geometrically exact shell model. Part III: Computational aspects of the nonlinear theory. Comput. Methods Appl. Mech. Eng. 79(1), 21–70 (1990)

    MATH  Google Scholar 

  62. Sprecher, M.: Numerical methods for optimization and variational problems with manifold-valued data. Ph.D. thesis, ETH Zürich (2016)

    Google Scholar 

  63. Stahl, S.: The Poincaré Half-Plane – A Gateway to Modern Geometry. Jones and Bartlett Publishers, Burlington (1993)

    MATH  Google Scholar 

  64. Struwe, M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60(1), 558–581 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  65. Walther, A., Griewank, A.: Getting started with ADOL-C. In: Naumann, U., Schenk, O. (eds.) Combinatorial Scientific Computing. Computational Science, pp. 181–202. Chapman-Hall CRC, Boca Raton (2012)

    Chapter  Google Scholar 

  66. Weinmann, A., Demaret, L., Storath, M.: Total variation regularization for manifold-valued data. SIAM J. Imaging Sci. 7(4), 2226–2257 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  67. Wriggers, P., Gruttmann, F.: Thin shells with finite rotations formulated in Biot stresses: theory and finite element formulation. Int. J. Numer. Methods Eng. 36, 2049–2071 (1993)

    Article  MATH  Google Scholar 

  68. Zeidler, E.: Nonlinear Functional Analysis and its Applications, vol. 1. Springer, New York (1986)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hanne Hardering .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Hardering, H., Sander, O. (2020). Geometric Finite Elements. In: Grohs, P., Holler, M., Weinmann, A. (eds) Handbook of Variational Methods for Nonlinear Geometric Data. Springer, Cham. https://doi.org/10.1007/978-3-030-31351-7_1

Download citation

Publish with us

Policies and ethics